The socle, lattice of normal subgroups, and faithful irreducible characters 1

A study of the Mobius function on the lattice of normal subgroups yields information of the number of conjugacy classes required to generate a finite group. It is shown how this number is reflected in the character table. A result of Geschutz states that faithful irreducible characters exist if and only if the socle is generated by a sole conjugacy class. By studying all faithful irreducible characters, generalizations of this result are presented. Faithful irreducible characters are shown to be orthogonal modulo the socle of the group. This leads, in particular, to information on the structure constants of the socle.